Parameterise the circle that results from intersecting the unit sphere in centered at the origin with the plane

I tried
however the last equation is an ellipse????

Jan 24th 2011, 12:43 PM

ahaok

Quote:

Originally Posted by FGT12

Parameterise the circle that results from intersecting the unit sphere in centered at the origin with the plane

I tried
however the last equation is an ellipse????

the intersection is an ellipse and no circle!(Wondering)

Jan 24th 2011, 07:23 PM

xxp9

You result is only the projection of the circle to the xy plane, which is really an ellipse.

Jan 25th 2011, 07:36 AM

FGT12

so how would you parameterise the circle in

Jan 25th 2011, 09:57 AM

Mazerakham

It's definitely going to be a circle, because really: how could a plane intersect a sphere and make an ellipse! You're right that it's absurd.

Have you tried drawing a sketch? Try visualizing the picture with the y axis pointing directly out of the paper, and that may give you some insight. I solved the problem by first coming up with an equation based on the sketch, then proving that it was right. So, like I said. Start with the picture.

Jan 25th 2011, 11:06 AM

HallsofIvy

ahaok is wrong, a sphere cut by a plane gives a circle, not an ellipse. xxp9 is right- a circle, projected onto a plane at an angle to a line perpendicular to the circle gives an ellipse. That is why you get the equation of an ellipse- you are projecting the circle onto the xy-plane, not parameterizing it.

Here, your sphere is the unit sphere, and the plane is given by x= z. The "standard" parameterization of the unit sphere can be derived from spherical coordinates:

by taking the radius variable, equal to 1:

The fact that z= x means that we must have so that .

So one way of getting the parameterization of the circle is to replace by , reducing from two parameters to one:

Of course, . And, we can use to say that and so have

.

Jan 25th 2011, 11:34 AM

TheEmptySet

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Quote:

Originally Posted by FGT12

Parameterise the circle that results from intersecting the unit sphere in centered at the origin with the plane

I tried
however the last equation is an ellipse????

The vector form for the plane is

This gives that Using the equation of the sphere gives

So a parametric representation of the circle is

Since the plus or minus is really messy it can be eliminated using trigonometry.